9 MEASURABLE FUNCTIONS 151
22. Let X be a locally compact space, Y a compact space, p a positive measure on X, and
(fn)nzo a sequence of /x-measurable mappings from X to Y. Show that there exists a
mapping G?, jc)i—>O-P(JC) of N x X into N with the following properties: (1) for each
integer p ^ 0 and each integer n J> 0, the set ctpl(n) is /u-measurable; (2) for each
x e X, the sequence (ftTp(x)(x))p^Q converges in Y. (One method is as follows. Taking
a distance function on Y, define universally measurable subsets As of Y satisfying
conditions (i) and (ii) of Section 4.2, Problem 3(a) and such that (in the same notation)
AS'nAS" = 0. Then define subsets Bs of X inductively as follows: if Bs has been
defined in such a way that, for each x e Bs, infinitely many terms of the sequence
(/„(*)) belong to A,, we define Bs- as the set of all points x e Bs such that infinitely
many terms of the sequence (fn(x)) belong to A,', and we define Bs« to be the com-
plement of BS' in Bs. Now let x e X and let (ep)p^0 be the sequence of O's and 1's
such that, if sp — (£t)o^t^P > then x e BSp for all p. Define crp(x) by induction as follows:
a0(x) is the smallest integer n such that/n(x) e AS(), and ap(x) is the smallest integer
n > op-i(x) such that fn(x) e A5p. Use Problem 21 to verify that this definition of ap(x)
satisfies the conditions of the problem.)
23. Let ju, be a bounded positive measure on X and let/5:; 0 be a /x-integrable function.
Show that there exists a lower semi continuous function g such that g^l/f (with
the convention that 1/0= -foo) and such that gf is integrable (with the convention
that the product of 0 and -f oo is 0). (Reduce to the case where/is bounded; consider
the sets An of points x e X such that f(x) g l/«, and apply (13.7.9) to these sets to
construct g.)
24. Let X, Y be locally compact spaces, p, a positive bounded measure on X, and TT a
/x-measurable mapping from X to Y. For each/e JTR(Y), show that /° TT is jii-integ-
rable and that the mapping /h-» (/o rr) dp, is a positive measure v on Y (called the
image of JJL under TT, and denoted by ^(/i)). Generalize the results of Section 13.8,
Problem 12, and Section 13.9, Problems 10 to 14.
25. Let X be a compact space and let (-07,,) be a sequence of finite partitions of X consisting
of integral sets, such that wm is finer then wn whenever m > n (Problem 7). An ele-
mentary martingale relative to the sequence (wn) is by definition a sequence (/„) of
integrable functions ^ 0 such that:
(i) /„ is constant on each set of the partition wn;
(ii) If m > n, for each set A e wn we have
(in other words, on each set A e w» the value of/„ is equal to the average of/m over A),
(a) Let <a, b be real numbers such that 0 < a < b, and let Eflb be the integrable set of
points x e X such that
lim inf fn(x) < a< b < lim sup/n(jc).
n-froo n-*oo
Show that Eab is negligible. (Suppose if possible that p,(Eab) > 0. For each integer
p > 0, let Fp be the union of the sets A e wn (n^p) such that A n Eab ^ 0; if F =
P) Fp, then ^(F) ^ p*(Eat)) > 0. Show that for each integer/? and each A e wn (n *Zp)
p
such that A n Efli, 7^ 0, we have b * ft(A n F^) ^ a - p,(A) for all q J> w, by using the exists a sequence (tn) of finite real numbers such that £ |fB| = -|- oo